If the physical geometry and the coordinate system selected permit the choice of a \emph{constant} ``normal'' direction designated by the unit vector $\hat{n}$ such the the transverse ($\vec{r}_\bot$) and normal ($n$) variables in Maxwell's equations can be separated such that the electric and magnetic fields can be expressed in the following form (see section \ref{sec:waveeqtransversenormal}),

TODO:  The following equations are not correct, I don't think in general you
can factor out $N(n)$ from the fields.  It may be that you can do this for the
vector potentials and in some cases the tangential fileds, but not the
totoal fields themselves. If you look at TEz and TMz fields the normal
components are different from the tangential components.
\begin{align}
	\vec{E}(\vec{r})&=[\vec{e}(\vec{r}_\bot)+e_n(\vec{r}_\bot)\hat{n}]V(n)\\ 
	\vec{H}(\vec{r})&=[\vec{h}(\vec{r}_\bot)+h_n(\vec{r}_\bot)\hat{n}]I(n)
\end{align}
then it is possible to rearrange Maxwell's equations to show that the transverse fields ($\vec{E}_\bot,\vec{H}_\bot$) can be expressed as a linear combination of two terms; one involving only the normal component $E_n$, and the other involving only the normal component $H_n$.  Because the two terms are added (i.e., superimposed), this rearrangement constitutes a demonstration that the two assumptions (1) $H_n=0$ or (2) $E_n=0$ lead to independent solutions (or modes) of Maxwell's equations.  For descriptive reasons, these two solutions are called respectively ``transverse magnetic modes'' (TM-modes) and ``transverse electric modes'' (TE-modes).

In Europe and other parts of the world, the common notation for these modes are ``E-modes'' and ``H-modes'' respectively because the first comes from $E_n$ alone and the second comes form $H_n$ alone.  This change in names can be a source of confusion when reading the EM (electromagnetic) literature so it is important to take note that ``TM-modes'' are ``E-modes'' and ``TE-modes'' are ``H-modes''. 

We usually use a subscript (or superscript) with these mode designators to indicate the normal direction.  For example, in the rectangular coordinate system if we use the z-direction as normal, then we would write $TM_z$ and $TE_z$ (or $TM^z$ and $TE^z$) to describe the TM- and TE-modes and the transverse fields ($\vec{E}_\bot, \vec{H}_\bot$) would lie in the x-y plane.  If the y-direction is selected as normal direction, we wold write $TM_y$ and $TE_y$ and the transverse fields would lie in the x-z plane, etc.

The final validation of separate TM- or TE-modes solutions to Maxwell's equations is whether or not all boundary conditions can be satisfied for the case selected. If all boundary conditions can be satisfied using $H_n=0$ and computing ($\vec{E}_\bot, \vec{H}_\bot$) from $E_n$ alone, then we are assured that TM$_n$-mode solutions exist.  In this case, TE$_n$-mode solutions will also be found to exist.  If both $H_n$ and $E_n$ field components are required to match all of the boundary conditions of a given geometry, then separate TM- and TE-modes do not exist, and we have a new case in which the elementary solutions are called ``hybrid-modes''.  This occurs, for example, in circular waveguide filled with multiple circular layers of dielectric material whose boundaries follow constant values of $\rho$.  In this case, one can still separate the longitudinal direction $n=z$, but the radial boundary conditions between layers cannot be satisfied using solutions based on $E_z$ alone or $H_z$ alone.  Both components are required which results in hybrid modes.

To show that Maxwell's equations permit TE and TM types of independent solutions of modes, we begin by dividing the vector operator and fields into transverse and normal components
\begin{align}
\vec{\nabla} &= \vec{\nabla}_\bot+\frac{\partial}{\partial n}\hat{n}\\
\vec{E}&= \vec{E}_\bot+E_n\hat{n}\\
\vec{H}&= \vec{H}_\bot+H_n\hat{n}
\end{align}
where the time variations are assumed sinusoidal and have been separated out.  With these separations, Maxwell's curl equations (\ref{eqn:flsfsep3}) and (\ref{eqn:alsfsep3})
\begin{align}
    \left(\vec{\nabla}_\bot+\frac{\partial}{\partial n}\hat{n}\right)\times\;(\vec{E}_\bot+{E}_n\hat{n})&=-j\omega\mu'(\vec{H}_\bot+{H}_n\hat{n})\label{eqn:flsfsep4}\\
    \left(\vec{\nabla}_\bot+\frac{\partial}{\partial n}\hat{n}\right)\times\,(\vec{H}_\bot+{H}_n\hat{n})&=j\omega\varepsilon'(\vec{E}_\bot+{E}_n\hat{n})\label{eqn:alsfsep4}
\end{align}
can be expanded and separated according to their transverse and normal directions to get the following four equations.  

1-D normal vector equations:
\begin{align}
\vec{\nabla}_\bot\times\vec{E}_\bot&=-j\omega\mu'{H}_n\hat{n}\label{eqn:onedimnormala}\\
\vec{\nabla}_\bot\times\vec{H}_\bot&=j\omega\varepsilon'{E}_n\hat{n}\label{eqn:onedimnormalb}
\end{align}

2-D transverse vector equations:
\begin{align}
\vec{\nabla}_\bot{E}_n\times\hat{n}+\hat{n}\times\frac{\partial\vec{E}_\bot}{\partial{n}}&=-j\omega\mu'\vec{H}_\bot\label{eqn:twodimtransversea}\\
\vec{\nabla}_\bot{H}_n\times\hat{n}+\hat{n}\times\frac{\partial\vec{H}_\bot}{\partial{n}}&=j\omega\varepsilon'\vec{E}_\bot\label{eqn:twodimtransverseb}
\end{align}
In equations (\ref{eqn:onedimnormala}) and (\ref{eqn:onedimnormalb}), we have used $\hat{n}\times{E_n}\hat{n}=0$ and $\hat{n}\times{H_n}\hat{n}=0$ which is true because the cross products involve vectors in the same direction.

We then proceed by using (\ref{eqn:twodimtransversea}) to eliminate $\vec{H}_\bot$ from (\ref{eqn:twodimtransverseb}), and using (\ref{eqn:twodimtransverseb}) to eliminate $\vec{E}_\bot$ from (\ref{eqn:twodimtransversea}).  Starting with (\ref{eqn:twodimtransverseb}) we get
\begin{align}
j\omega\varepsilon'\vec{E}_\bot&=\vec{\nabla}_\bot{H}_n\times\hat{n}+\hat{n}\times\frac{\partial}{\partial{n}}\left[\frac{1}{-j\omega\mu'}\left(\vec{\nabla}_\bot{E}_n\times\hat{n}+\hat{n}\times\frac{\partial\vec{E}_\bot}{\partial{n}}\right)\right]
\end{align}
and then rearranging to get, 
\begin{align}
j\omega\varepsilon'\vec{E}_\bot&=\vec{\nabla}_\bot{H}_n\times\hat{n}+\frac{1}{-j\omega\mu'}\left[\hat{n}\times\left(\vec{\nabla}_\bot\frac{\partial{E}_n}{\partial{n}}\times\hat{n}\right)+\hat{n}\times\left(\hat{n}\times\frac{\partial^2\vec{E}_\bot}{\partial{n}^2}\right)\right]\label{eqn:TETMmodeSolntemp1}
\end{align}
If we now apply the vector identities
\begin{align}
\hat{n}\times(\vec{\nabla}_\bot{A}_n\times\hat{n})&=\vec{\nabla}_\bot\vec{A}_\bot(\hat{n}\cdot\hat{n})-\underbrace{(\hat{n}\cdot\vec{\nabla}_\bot)\vec{A}_n}_{=0 \;\text{orthogonal}}=\vec{\nabla}_\bot\vec{A}_\bot\\
\hat{n}\times(\hat{n}\times\vec{A}_\bot)&=\underbrace{(\hat{n}\cdot\vec{A}_\bot)\hat{n}}_{=0\;\text{orthogonal}}-(\hat{n}\cdot\hat{n})\vec{A}_\bot=-\vec{A}_\bot
\end{align}
to (\ref{eqn:TETMmodeSolntemp1}) yields,
\begin{align}
j\omega\varepsilon'\vec{E}_\bot&=\vec{\nabla}_\bot{H}_n\times\hat{n}+\frac{1}{-j\omega\mu'}\left[\vec{\nabla}_\bot\frac{\partial{E}_n}{\partial{n}}-\frac{\partial^2\vec{E}_\bot}{\partial{n}^2}\right]
\end{align}
By rearranging terms to put the transverse field components on the left, we get
\begin{align}
\left(\frac{\partial^2}{\partial{n}^2}+k^2\right)\vec{E}_\bot=\vec{\nabla}_\bot\frac{\partial{E}_n}{\partial{n}}-j\omega\mu'\vec{\nabla}_\bot{H}_n\times\hat{n}\label{eqn:TETMmodeSolntemp2}
\end{align}
where $k^2$ is defined in (\ref{eqn:gk1}).  The same operations on (\ref{eqn:twodimtransversea}) gives
\begin{align}
\left(\frac{\partial^2}{\partial{n}^2}+k^2\right)\vec{H}_\bot=j\omega\varepsilon'\vec{\nabla}_\bot{E}_n\times\hat{n}+\vec{\nabla}_\bot\frac{\partial{H}_n}{\partial{n}}\label{eqn:TETMmodeSolntemp3}
\end{align}
Substituting (\ref{eqn:constrainteqn2d}) and (\ref{eqn:TETMmodeSolntemp2}) into (\ref{eqn:TETMmodeSolntemp3})  yields,
\begin{align}
\left(\frac{\partial^2}{\partial{n}^2}+k_n^2\right)\vec{E}_\bot+k_\bot^2\vec{E}_\bot&=\vec{\nabla}_\bot\frac{\partial{E}_n}{\partial{n}}-j\omega\mu'\vec{\nabla}_\bot{H}_n\times\hat{n}\label{eqn:TETMmodeSolntemp4a}\\
\left(\frac{\partial^2}{\partial{n}^2}+k_n^2\right)\vec{H}_\bot+k_\bot^2\vec{H}_\bot&=j\omega\varepsilon'\vec{\nabla}_\bot{E}_n\times\hat{n}+\vec{\nabla}_\bot\frac{\partial{H}_n}{\partial{n}}\label{eqn:TETMmodeSolntemp4b}
\end{align}
and then using (\ref{eqn:scalarwavenorm}) and solving for the transverse fields gives,
\begin{align}
\vec{E}_\bot=\frac{1}{k_\bot^2}\left[\vec{\nabla}_\bot\frac{\partial{E}_n}{\partial{n}}-j\omega\mu'\vec{\nabla}_\bot{H}_n\times\hat{n}\right]\label{eqn:TETMmodeSolntemp4}\\
\vec{H}_\bot=\frac{1}{k_\bot^2}\left[j\omega\varepsilon'\vec{\nabla}_\bot{E}_n\times\hat{n}+\vec{\nabla}_\bot\frac{\partial{H}_n}{\partial{n}}\right]\label{eqn:TETMmodeSolntemp5}
\end{align}

Equations (\ref{eqn:TETMmodeSolntemp4}) and (\ref{eqn:TETMmodeSolntemp5}) give the fields perpendicular to the normal direction in terms of the normal field components ($E_n$, $H_n$).  Each of the right hand sides is a sum (i.e., a superposition) of two types of fields which we designate as TM$_n$ (those fields which come form $E_n$ alone) and TE$_n$ (those fields which come form $H_n$ alone).  Whenever these two types of fields can stand alone and satisfy all the geometric boundary conditions, then the two types of fields constitute independent solutions to Maxwell's equations; i.e., TM$_n$-modes and TE$_n$-modes.

Thus a complete field solution would come by adding
\begin{align}
\vec{E}_\bot^\text{TM} &= \frac{1}{k_\bot^2}\vec{\nabla}_\bot\frac{\partial{E}_n}{\partial{n}}\\
\vec{H}_\bot^\text{TM} &= \frac{j\omega\varepsilon'}{k_\bot^2}\vec{\nabla}_\bot{E}_n\times\hat{n}
\end{align}
and
\begin{align}
\vec{E}_\bot^\text{TE} &= \frac{-j\omega\mu'}{k_\bot^2}\vec{\nabla}_\bot{H}_n\times\hat{n}\\
\vec{H}_\bot^\text{TE} &= \frac{1}{k_\bot^2}\vec{\nabla}_\bot\frac{\partial{H}_n}{\partial{n}}
\end{align}
to get
\begin{align}
\vec{E}_\bot&=\vec{E}_\bot^\text{TM}+\vec{E}_\bot^\text{TE} \\
\vec{H}_\bot&=\vec{H}_\bot^\text{TM}+\vec{H}_\bot^\text{TE}
\end{align}

We conclude by noting that in some cases such as two wire transmission line fields, both $E_n$ and $H_n$ can be zero.  In this case, solutions to Maxwell's equations (in terms of $\vec{E}_\bot$ and $\vec{H}_\bot$ fields) are still possible using vector potential theory and the resulting modes are called TEM-modes since all fields are transverse to the normal direction.  Vector potential theory shows that in this case $k_\bot=0$ and (\ref{eqn:TETMmodeSolntemp4}) and (\ref{eqn:TETMmodeSolntemp5}) cannot be used.
